Lecture 2: Risk and Return
Measuring Assets’ Returns

- Let’s begin by focusing on two major asset classes - stocks and
bonds
- How have these assets performed historically?
- What’s the best way to summarize their risk + return?
Measuring Assets’ Returns

- Why does a Treasury bill’s return not fall below zero?
- Is monthly return the best measure? What else could we do?
- Let’s quickly refresh ourselves on how to measure returns.
Return Definitions
- The rate of return is also known as the “net return”
\[
r_{t} = \frac{P_{t} + D_{t}}{P_{t-1}} - 1
\]
- Gross returns (sometimes \(R_{t}\)) is referred to as \(1+r_{t}\)
- The risk-free rate will be called \(r_{f}\)
- How can I get the risk-free rate?
- Excess returns above the risk free rate are \(r_{e} = r - r_{f}\)
Return Defintions
A holding-period return of \(T\)
years is
\[
r_{0,T} = (1+r_{1})(1+r_{2})\ldots (1+r_{T}) - 1
\]
- How you measure the holding period matters a ton!
- Recall the mutual fund experiment

Return Definitions
- How do we compare returns across different horizons?
- Say I want to compare a window of cumulative returns 5 v 10
years.
- Annualized returns on the cumulative return: \[
\widetilde{r}_{0,T} = (1+r_{0,T})^{\frac{1}{T_{\text{years}}}} - 1
\]

Return Definitions
- Arithmetic average returns \[
\overline{r} = \frac{1}{T}(r_{1} + r_{2} + \ldots + r_{T})
\]
- Geometric average returns \[
r_{G} = \left[(1+r_{1})(1+r_{2})\ldots (1+r_{T})\right]^{\frac{1}{T}} -
1 = \big[\frac{P_{T}}{P_{0}}\big]^{\frac{1}{T}} - 1
\]
- Arithmetic average is unbiased estimate of 1-period future returns
(expected returns)
- Geometric average is a measure of cumulative past performance
Return Definitions
- We calculate estimates of variance using squared deviations from
arithmetic average returns \[
\sigma^{2}(r) = VAR(r) = \frac{1}{T}\bigg((r_{1}-\overline{r})^{2} +
(r_{2}-\overline{r})^{2} + \ldots + (r_{T}-\overline{r})^{2})\bigg)
\]
- Standard deviation is the square-root of variance: \[
\sigma(r) = SD(r) = \sqrt{\frac{1}{T}\bigg((r_{1}-\overline{r})^{2} +
(r_{2}-\overline{r})^{2} + \ldots + (r_{T}-\overline{r})^{2})\bigg)}
\]
Return Defintions
- Finally, covariance measures how two returns move
together \[
\begin{aligned}
\sigma_{i,j} = COV(r_{i}, r_{j}) =
\frac{1}{T}\bigg((r_{1,i}-\overline{r}_{i})(r_{1,j}-\overline{r}_{j})
&+ (r_{2,i}-\overline{r}_{i})(r_{2,j}-\overline{r}_{j})\\
&+ \ldots +
(r_{T,i}-\overline{r}_{i})(r_{T,j}-\overline{r}_{j})\bigg)
\end{aligned}
\]
- Correlations scale the covariance by standard deviations \[
\rho_{i,j} = \frac{\sigma_{i,j}}{\sigma_{i}\sigma_{j}}
\]
- Excel provides functions AVERAGE,GEOMEAN,VAR, STDEV, and COVAR
- To make investment decisions, we need to know
- … the expected future returns
- … the riskiness of future returns
- We turn to historical return data for these
- A caveat…
- The data give a pretty good sense of the risk, but expected returns
are hard to measure.
- Why?
Mean 1 (purple): 4.03
Mean 2 (yellow): 4.69
True distributions:
Mean 1: 4
Mean 2: 5
Asset Returns: Historical
Record

- US returns high, but not an outlier
Asset Returns: Historical
Record

- Average s.d. of returns is 23% (US 20%)
- What’s up with Italy, Germany, Japan?
High
returns or survivorship bias? (Jorion and Goetzmann (1999))
High
returns or survivorship bias? (Anarkulova et al. 2020)
“From 1990 to 2019, a diversified investment in Japanese stocks
produced returns of -9% in nominal terms” Anarkulova et
al. 2020
Incorporating the losses from buy-and-hold strategies that were
selected away.
Still waiting for data?
Most recent U.S. historical data suggest:
- Average excess returns of large stocks over long term bonds (\(r_{e}\)) of roughly 6% … with a standard
deviation of 20% (\(\sigma_{e}\))
- With 81 years of data, standard error associated
with mean excess returns is \[
\sigma(\overline{r}_{e}) = \frac{\sigma(r_{e})}{\sqrt{T}} = \frac{20
\%}{9} \approx 2.2\%
\]
- Can we reject the null hypothesis that excess returns are 4%… or
2%?
How can we forecast
returns going forward?
Recall that
\[
r_{1} = \underbrace{\frac{D_{1}}{P_{0}}}_{\text{dividend yield}} +
\underbrace{\frac{P_{1}}{P_{0}}}_{\text{growth}}
\]
Also,
\[
\underbrace{\frac{D_{1}}{P_{0}}}_{\text{dividend yield}} =
\text{earnings payout share} \times \frac{E_{1}}{P_{0}}
\] where, \(E_{1}\) is
earnings.
This implies \[
r_{1} = \text{firm's payout share} \times \frac{E_{1}}{P_{0}} +
\text{growth}
\]
The Gordon Model
\[
r_{1} = \text{payout share} \times \frac{E_{1}}{P_{0}} + \text{growth}
\]
- Historically, payout has been around 50%, P/E about 25 and growth
rate of about 4-5% (nominal)
- What does that imply about nominal expected returns?
- Estimate is about 6-7% nominal expected return for stocks, or 4%
real, or a risk premium of 2-3% over nominal bonds
- This is low versus history
- To believe anything else, you must disagree with payout, E/P, or
growth.
Shiller Price-Earnings
Ratios

Shiller Price-Earnings
Ratios

Other risk measures
Value at Risk (VaR)
- A measure of loss most frequently associated with extreme negative
returns
- VaR is the quantile of a distribution below which lies q % of the
possible values of that distribution
- The 5% VaR, commonly estimated in practice, tells us how bad returns
(or losses) will be in the worst 5% of times
VaR under normality
- 5% VaR return is equal to E(r) - 1.645 X s.d.
- 5% Value at Risk (VaR) represents the lower bound on the return’s
10% confidence interval
- Under non-normality, look at historic returns for 5% cutoff

Expected Shortfall (ES)
- Also called conditional tail expectation (CTE)
- More conservative measure of downside risk than VaR
- VaR takes the highest return from the worst cases
- ES takes an average return of the worst cases

How would you want to model risk?
Consider the Swiss Franc
case:
Takeaways
- We still have a limited amount of high quality data to make
inferences from, but…
- Historical data suggests a premium for risk in asset returns, but
the size of the premium is up for debate
- If expected returns for the aggregate stock market
with 80+ years of data are so imprecise, what are we to do with
individual stocks?
- Need more than data to understand expected returns